nprmdvdsfacm1lem4 - Mathbox for Alexander van der Vekens

	Mathbox for Alexander van der Vekens		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > nprmdvdsfacm1lem4		Structured version Visualization version GIF version

Theorem nprmdvdsfacm1lem4 48342

Description: Lemma 4 for nprmdvdsfacm1 48343. (Contributed by AV, 7-Apr-2026.)

**Assertion**
Ref	Expression
nprmdvdsfacm1lem4	⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → 𝑁 ∥ (!‘(𝑁 − 1)))

**Proof of Theorem nprmdvdsfacm1lem4**
Step	Hyp	Ref	Expression
1		eluzelz 12871	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘6) → 𝑁 ∈ ℤ)
2	1	3ad2ant1 1149	. 2 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → 𝑁 ∈ ℤ)
3		elfzoelz 13687	. . . 4 ⊢ (𝐴 ∈ (2..^𝑁) → 𝐴 ∈ ℤ)
4		id 23	. . . . 5 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℤ)
5		2z 12625	. . . . . . 7 ⊢ 2 ∈ ℤ
6	5	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ ℤ → 2 ∈ ℤ)
7	6, 4	zmulcld 12705	. . . . 5 ⊢ (𝐴 ∈ ℤ → (2 · 𝐴) ∈ ℤ)
8	4, 7	zmulcld 12705	. . . 4 ⊢ (𝐴 ∈ ℤ → (𝐴 · (2 · 𝐴)) ∈ ℤ)
9	3, 8	syl 18	. . 3 ⊢ (𝐴 ∈ (2..^𝑁) → (𝐴 · (2 · 𝐴)) ∈ ℤ)
10	9	3ad2ant2 1150	. 2 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → (𝐴 · (2 · 𝐴)) ∈ ℤ)
11		1z 12623	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
12		6nn 12329	. . . . . . . . . . 11 ⊢ 6 ∈ ℕ
13	12	nnzi 12617	. . . . . . . . . 10 ⊢ 6 ∈ ℤ
14		1re 11207	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
15		6re 12330	. . . . . . . . . . 11 ⊢ 6 ∈ ℝ
16		1lt6 12427	. . . . . . . . . . 11 ⊢ 1 < 6
17	14, 15, 16	ltleii 11332	. . . . . . . . . 10 ⊢ 1 ≤ 6
18		eluz2 12867	. . . . . . . . . 10 ⊢ (6 ∈ (ℤ_≥‘1) ↔ (1 ∈ ℤ ∧ 6 ∈ ℤ ∧ 1 ≤ 6))
19	11, 13, 17, 18	mpbir3an 1358	. . . . . . . . 9 ⊢ 6 ∈ (ℤ_≥‘1)
20		uzss 12884	. . . . . . . . 9 ⊢ (6 ∈ (ℤ_≥‘1) → (ℤ_≥‘6) ⊆ (ℤ_≥‘1))
21	19, 20	ax-mp 5	. . . . . . . 8 ⊢ (ℤ_≥‘6) ⊆ (ℤ_≥‘1)
22	21	sseli 3932	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘6) → 𝑁 ∈ (ℤ_≥‘1))
23		elnnuz 12901	. . . . . . 7 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ_≥‘1))
24	22, 23	sylibr 237	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘6) → 𝑁 ∈ ℕ)
25	24	3ad2ant1 1149	. . . . 5 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → 𝑁 ∈ ℕ)
26		nnm1nn0 12544	. . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
27	25, 26	syl 18	. . . 4 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → (𝑁 − 1) ∈ ℕ₀)
28	27	faccld 14320	. . 3 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → (!‘(𝑁 − 1)) ∈ ℕ)
29	28	nnzd 12616	. 2 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → (!‘(𝑁 − 1)) ∈ ℤ)
30		nprmdvdsfacm1lem1 48339	. 2 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → 𝑁 ∥ (𝐴 · (2 · 𝐴)))
31		elfzo2 13690	. . . . . 6 ⊢ (𝐴 ∈ (2..^𝑁) ↔ (𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝐴 < 𝑁))
32		2eluzge1 12905	. . . . . . . . . 10 ⊢ 2 ∈ (ℤ_≥‘1)
33		uzss 12884	. . . . . . . . . 10 ⊢ (2 ∈ (ℤ_≥‘1) → (ℤ_≥‘2) ⊆ (ℤ_≥‘1))
34	32, 33	ax-mp 5	. . . . . . . . 9 ⊢ (ℤ_≥‘2) ⊆ (ℤ_≥‘1)
35	34	sseli 3932	. . . . . . . 8 ⊢ (𝐴 ∈ (ℤ_≥‘2) → 𝐴 ∈ (ℤ_≥‘1))
36	35	3ad2ant1 1149	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝐴 < 𝑁) → 𝐴 ∈ (ℤ_≥‘1))
37	5	a1i 11	. . . . . . . . 9 ⊢ (𝐴 ∈ (ℤ_≥‘2) → 2 ∈ ℤ)
38		eluzelz 12871	. . . . . . . . 9 ⊢ (𝐴 ∈ (ℤ_≥‘2) → 𝐴 ∈ ℤ)
39	37, 38	zmulcld 12705	. . . . . . . 8 ⊢ (𝐴 ∈ (ℤ_≥‘2) → (2 · 𝐴) ∈ ℤ)
40	39	3ad2ant1 1149	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝐴 < 𝑁) → (2 · 𝐴) ∈ ℤ)
41		1lt2 12412	. . . . . . . 8 ⊢ 1 < 2
42		eluz2 12867	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (ℤ_≥‘2) ↔ (2 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 2 ≤ 𝐴))
43		zre 12594	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
44	43	3ad2ant2 1150	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 2 ≤ 𝐴) → 𝐴 ∈ ℝ)
45		2re 12314	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ
46	45	a1i 11	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 2 ≤ 𝐴) → 2 ∈ ℝ)
47		2pos 12344	. . . . . . . . . . . . . . 15 ⊢ 0 < 2
48		0red 11210	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℤ → 0 ∈ ℝ)
49	45	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℤ → 2 ∈ ℝ)
50	48, 49, 43	3jca 1144	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℤ → (0 ∈ ℝ ∧ 2 ∈ ℝ ∧ 𝐴 ∈ ℝ))
51		ltletr 11301	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℝ ∧ 2 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((0 < 2 ∧ 2 ≤ 𝐴) → 0 < 𝐴))
52	50, 51	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℤ → ((0 < 2 ∧ 2 ≤ 𝐴) → 0 < 𝐴))
53	47, 52	mpani 708	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℤ → (2 ≤ 𝐴 → 0 < 𝐴))
54	53	imp 411	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 2 ≤ 𝐴) → 0 < 𝐴)
55	54	3adant1 1146	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 2 ≤ 𝐴) → 0 < 𝐴)
56	44, 46, 55	3jca 1144	. . . . . . . . . . 11 ⊢ ((2 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 2 ≤ 𝐴) → (𝐴 ∈ ℝ ∧ 2 ∈ ℝ ∧ 0 < 𝐴))
57	42, 56	sylbi 220	. . . . . . . . . 10 ⊢ (𝐴 ∈ (ℤ_≥‘2) → (𝐴 ∈ ℝ ∧ 2 ∈ ℝ ∧ 0 < 𝐴))
58	57	3ad2ant1 1149	. . . . . . . . 9 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝐴 < 𝑁) → (𝐴 ∈ ℝ ∧ 2 ∈ ℝ ∧ 0 < 𝐴))
59		ltmulgt12 12074	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 2 ∈ ℝ ∧ 0 < 𝐴) → (1 < 2 ↔ 𝐴 < (2 · 𝐴)))
60	58, 59	syl 18	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝐴 < 𝑁) → (1 < 2 ↔ 𝐴 < (2 · 𝐴)))
61	41, 60	mpbii 236	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝐴 < 𝑁) → 𝐴 < (2 · 𝐴))
62	36, 40, 61	3jca 1144	. . . . . 6 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝐴 < 𝑁) → (𝐴 ∈ (ℤ_≥‘1) ∧ (2 · 𝐴) ∈ ℤ ∧ 𝐴 < (2 · 𝐴)))
63	31, 62	sylbi 220	. . . . 5 ⊢ (𝐴 ∈ (2..^𝑁) → (𝐴 ∈ (ℤ_≥‘1) ∧ (2 · 𝐴) ∈ ℤ ∧ 𝐴 < (2 · 𝐴)))
64		elfzo2 13690	. . . . 5 ⊢ (𝐴 ∈ (1..^(2 · 𝐴)) ↔ (𝐴 ∈ (ℤ_≥‘1) ∧ (2 · 𝐴) ∈ ℤ ∧ 𝐴 < (2 · 𝐴)))
65	63, 64	sylibr 237	. . . 4 ⊢ (𝐴 ∈ (2..^𝑁) → 𝐴 ∈ (1..^(2 · 𝐴)))
66	65	3ad2ant2 1150	. . 3 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → 𝐴 ∈ (1..^(2 · 𝐴)))
67	11	a1i 11	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → 1 ∈ ℤ)
68	5	a1i 11	. . . . . . . 8 ⊢ (𝐴 ∈ (2..^𝑁) → 2 ∈ ℤ)
69	68, 3	zmulcld 12705	. . . . . . 7 ⊢ (𝐴 ∈ (2..^𝑁) → (2 · 𝐴) ∈ ℤ)
70	69	3ad2ant2 1150	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → (2 · 𝐴) ∈ ℤ)
71	47	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℤ → 0 < 2)
72		lemul2 12067	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (2 ≤ 𝐴 ↔ (2 · 2) ≤ (2 · 𝐴)))
73	49, 43, 49, 71, 72	syl112anc 1399	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℤ → (2 ≤ 𝐴 ↔ (2 · 2) ≤ (2 · 𝐴)))
74		1red 11208	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (2 · 2) ≤ (2 · 𝐴)) → 1 ∈ ℝ)
75		2t2e4 12403	. . . . . . . . . . . . . . 15 ⊢ (2 · 2) = 4
76		4re 12324	. . . . . . . . . . . . . . 15 ⊢ 4 ∈ ℝ
77	75, 76	eqeltri 2857	. . . . . . . . . . . . . 14 ⊢ (2 · 2) ∈ ℝ
78	77	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (2 · 2) ≤ (2 · 𝐴)) → (2 · 2) ∈ ℝ)
79	7	zred 12699	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℤ → (2 · 𝐴) ∈ ℝ)
80	79	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (2 · 2) ≤ (2 · 𝐴)) → (2 · 𝐴) ∈ ℝ)
81		1lt4 12418	. . . . . . . . . . . . . . . 16 ⊢ 1 < 4
82	14, 76, 81	ltleii 11332	. . . . . . . . . . . . . . 15 ⊢ 1 ≤ 4
83	82, 75	breqtrri 5137	. . . . . . . . . . . . . 14 ⊢ 1 ≤ (2 · 2)
84	83	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (2 · 2) ≤ (2 · 𝐴)) → 1 ≤ (2 · 2))
85		simpr 489	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ (2 · 2) ≤ (2 · 𝐴)) → (2 · 2) ≤ (2 · 𝐴))
86	74, 78, 80, 84, 85	letrd 11366	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ (2 · 2) ≤ (2 · 𝐴)) → 1 ≤ (2 · 𝐴))
87	73, 86	sylbida 603	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 2 ≤ 𝐴) → 1 ≤ (2 · 𝐴))
88	87	3adant1 1146	. . . . . . . . . 10 ⊢ ((2 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 2 ≤ 𝐴) → 1 ≤ (2 · 𝐴))
89	42, 88	sylbi 220	. . . . . . . . 9 ⊢ (𝐴 ∈ (ℤ_≥‘2) → 1 ≤ (2 · 𝐴))
90	89	3ad2ant1 1149	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝐴 < 𝑁) → 1 ≤ (2 · 𝐴))
91	31, 90	sylbi 220	. . . . . . 7 ⊢ (𝐴 ∈ (2..^𝑁) → 1 ≤ (2 · 𝐴))
92	91	3ad2ant2 1150	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → 1 ≤ (2 · 𝐴))
93		eluz2 12867	. . . . . 6 ⊢ ((2 · 𝐴) ∈ (ℤ_≥‘1) ↔ (1 ∈ ℤ ∧ (2 · 𝐴) ∈ ℤ ∧ 1 ≤ (2 · 𝐴)))
94	67, 70, 92, 93	syl3anbrc 1360	. . . . 5 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → (2 · 𝐴) ∈ (ℤ_≥‘1))
95		zsqcl 14165	. . . . . . 7 ⊢ (𝐴 ∈ ℤ → (𝐴↑2) ∈ ℤ)
96	3, 95	syl 18	. . . . . 6 ⊢ (𝐴 ∈ (2..^𝑁) → (𝐴↑2) ∈ ℤ)
97	96	3ad2ant2 1150	. . . . 5 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → (𝐴↑2) ∈ ℤ)
98		3z 12626	. . . . . . . 8 ⊢ 3 ∈ ℤ
99	98	a1i 11	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → 3 ∈ ℤ)
100	3	3ad2ant2 1150	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → 𝐴 ∈ ℤ)
101		nprmdvdsfacm1lem2 48340	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → 3 ≤ 𝐴)
102		eluz2 12867	. . . . . . 7 ⊢ (𝐴 ∈ (ℤ_≥‘3) ↔ (3 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 3 ≤ 𝐴))
103	99, 100, 101, 102	syl3anbrc 1360	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → 𝐴 ∈ (ℤ_≥‘3))
104		2timesltsq 48082	. . . . . 6 ⊢ (𝐴 ∈ (ℤ_≥‘3) → (2 · 𝐴) < (𝐴↑2))
105	103, 104	syl 18	. . . . 5 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → (2 · 𝐴) < (𝐴↑2))
106	94, 97, 105	elfzod 13691	. . . 4 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → (2 · 𝐴) ∈ (1..^(𝐴↑2)))
107		oveq2 7418	. . . . . 6 ⊢ (𝑁 = (𝐴↑2) → (1..^𝑁) = (1..^(𝐴↑2)))
108	107	eleq2d 2847	. . . . 5 ⊢ (𝑁 = (𝐴↑2) → ((2 · 𝐴) ∈ (1..^𝑁) ↔ (2 · 𝐴) ∈ (1..^(𝐴↑2))))
109	108	3ad2ant3 1151	. . . 4 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → ((2 · 𝐴) ∈ (1..^𝑁) ↔ (2 · 𝐴) ∈ (1..^(𝐴↑2))))
110	106, 109	mpbird 260	. . 3 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → (2 · 𝐴) ∈ (1..^𝑁))
111		muldvdsfacm1 48091	. . 3 ⊢ ((𝐴 ∈ (1..^(2 · 𝐴)) ∧ (2 · 𝐴) ∈ (1..^𝑁)) → (𝐴 · (2 · 𝐴)) ∥ (!‘(𝑁 − 1)))
112	66, 110, 111	syl2anc 595	. 2 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → (𝐴 · (2 · 𝐴)) ∥ (!‘(𝑁 − 1)))
113	2, 10, 29, 30, 112	dvdstrd 16352	1 ⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → 𝑁 ∥ (!‘(𝑁 − 1)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1101 = wceq 1568 ∈ wcel 2141 ⊆ wss 3904 class class class wbr 5108 ‘cfv 6536 (class class class)co 7410 ℝcr 11098 0cc0 11099 1c1 11100 · cmul 11104 < clt 11242 ≤ cle 11243 − cmin 11440 ℕcn 12232 2c2 12294 3c3 12295 4c4 12296 6c6 12298 ℕ₀cn0 12503 ℤcz 12590 ℤ_≥cuz 12861 ..^cfzo 13682 ↑cexp 14097 !cfa 14309 ∥ cdvds 16309

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5256 ax-nul 5268 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2095 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7862 df-1st 7985 df-2nd 7986 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-er 8693 df-en 8943 df-dom 8944 df-sdom 8945 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-div 11871 df-nn 12233 df-2 12302 df-3 12303 df-4 12304 df-5 12305 df-6 12306 df-n0 12504 df-z 12591 df-uz 12862 df-rp 13016 df-fz 13535 df-fzo 13683 df-seq 14038 df-exp 14098 df-fac 14310 df-bc 14339 df-dvds 16310

This theorem is referenced by: nprmdvdsfacm1 48343

W3C validator